ITS AN OPTIMIZATION QUESTION( LINEAR PROGRAMMING) You and several friends are about to prepare a lasagna dinner. The tasks to be performed, their predecessors and their estimated durations are below.

(a) Draw a project network for this problem.

(b) Find a forward numbering of the vertices of the network.

(c) Find the length si of a longest directed path from the start vertex to every other vertex.

(d) By reversing all arcs (and reversing the forward numbering) find the length ti of a longest directed path in the original network from each vertex i to the finish vertex.

(e) Give a schedule which completes all the activities in a minimum amount of time.

(f) What is the minimum amount of time required to complete all the activities?__________ Justify your answer. (There should be two parts to your justification.)

(g) For each activity (i,k), give the earliest starting time, the earliest finish time, the latest start time, the latest finish time, and the slack time of the activity, and state whether it is a critical activity. (Latest start and finish time mean latest time the activity can start / finish without delaying the completion time of the dinner.)

(h) Because of a phone call, you were interrupted for 6 minutes when you should have been cutting the onions and mushrooms. By how much will the dinner be delayed? If you use your food processor, which reduced the cutting time from 7 to 2 minutes, will the dinner still be delayed?

task	prodecessor	time
A-Buy the mozzarella cheese	none	30
B-Slice the mozzarella cheese	A	5
C-Beat 2 eggs	None	2
D-Mix eggs and ricotta cheese	C	3
E-Cut up onions and mushrooms	none	7
F-Cook the tomato sauce	E	25
G-Boil large quantity of water	none	15
H-Boil the lasagna noodles	G	10
I-Drain the lasagna noodles	H	2
J-Assemble all ingredients	B, D, F, I	10
K-Preheat the oven-	none	15
L-Bake the lasagna	J, K	30